## Introduction

Magnetic reconnection converts magnetic energy into plasma thermal and kinetic energy in laboratory, space, and astrophysical plasmas. Two major and largely separate endeavors have been pursued to quantitatively predict how the plasma is energized by reconnection1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16 and the rate at which reconnection proceeds17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38. Nevertheless, the linkage between these two fundamental aspects of reconnection is missing. The most critical question in understanding fast reconnection is what localizes the diffusion region (DR)39 (i.e., what makes it far shorter than the system size), giving rise to an open geometry of reconnection outflow. Petschek’s model19 provides a valid steady-state solution for such an open outflow geometry, but it fails to provide a valid localization mechanism in the uniform resistivity magnetohydrodynamics (MHD) model40,41; reconnection in such system always results in a system-size long diffusion region, known as the Sweet-Parker solution17,18. The idea of a spatially localized anomalous resistivity was later invoked to explain the localization needed42,43,44, but no clear evidence of such anomalous resistivity during collisionless reconnection has yet been identified.

Kinetic simulations beyond the MHD model suggest that antiparallel reconnection with an open outflow geometry occurs when the current sheet thins down to the ion inertial scale45,46,47,48. When this occurs, the Hall term in the generalized Ohm’s law49,50 dominates the electric field in the ion diffusion region (IDR), where the ions become demagnetized. The correlation between the Hall effect and fast reconnection was clearly demonstrated in the GEM reconnection challenge study21; this study showed that simulation models with the Hall term in the generalized Ohm’s law (particle-in-cell (PIC), hybrid, and Hall-MHD) realize fast reconnection, while only the uniform resistive-MHD model, which lacks the Hall term, exhibits a slow rate17,18. However, it remains unclear why and how the Hall term localizes the diffusion region, producing an open geometry. The dispersive property of waves arising from the Hall term was proposed as an explanation22,51,52,53, but the role of dispersive waves derived from linear analysis was called into question because reconnection can be fast even in systems that lack dispersive waves23,26,27,28.

In this work, we illustrate the role of Hall physics in plasma energization and why this causes the open geometry necessary to achieve fast reconnection in electron-ion plasmas. The two key points are: (1) the Hall term EHall = J × B/nec, while it dominates the electric field within the IDR, does not convert energy into plasmas because JEHall = J (J × B)/nec = 0. Thus, the inflowing plasma only gains a small amount of thermal energy within the IDR. (2) Insufficient pressure buildup at the x-line, where the magnetic field lines change their connectivity, causes the upstream magnetic pressure to locally pinch the diffusion region, opening out the exhaust54. For reconnection of antiparallel magnetic fields, an open geometry occurs if $$P{| }_{{{{{{{{\rm{xline}}}}}}}}} < {B}_{{{{{{\mathrm{x}}}}}}0}^{2}/8{{{{{{{\rm{\pi }}}}}}}}+{P}_{0}$$, where Pxline is the thermal pressure at the x-line, P0 is the asymptotic thermal pressure and $${B}_{{{{{{\mathrm{x}}}}}}0}^{2}/8{{{{{{{\rm{\pi }}}}}}}}$$ is the magnetic pressure based on the asymptotic magnetic field Bx0 far upstream from the IDR. These two results are used to derive a first-principles theory of the reconnection rate (the phrase “first-principles” refers to a theory that does not rely on measured empirical inputs from the simulations or observations). In order to show this pressure depletion during magnetic reconnection in electron-ion plasmas, we use PIC simulations to investigate the role of Hall electromagnetic fields in energy conversion and kinetic heating near the x-line. The cross-scale coupling from the mesoscale upstream MHD region, the IDR, and down to the electron diffusion region (EDR) is treated to obtain a prediction of the reconnection rate. Finally, we extend the discussion to systems without the Hall term, including electron-positron (pair) plasmas and resistive-MHD reconnection, explaining why the former is fast while the latter does not have an open outflow and is slow. We show that the same theoretical approach leads to the Sweet-Parker scaling, and provides the reason of why Sweet-Parker reconnection has a system-size long diffusion region.

## Results

We use 2-D PIC simulations to illustrate the key features of energy conversion in the diffusion region. Details of the simulation setup are in the “Methods” section. The units used in the presentation include the ion cyclotron time $${{{\Omega }}}_{{{{{{{{\rm{ci}}}}}}}}}^{-1}\,\equiv \,{({{{{{{{\rm{e}}}}}}}}{B}_{{{{{{\mathrm{x}}}}}}0}/{m}_{{{{{{{{\rm{i}}}}}}}}}c)}^{-1}$$, the Alfvén speed $${V}_{{{{{{{{\rm{A0}}}}}}}}}\,\equiv \,{B}_{{{{{{\mathrm{x}}}}}}0}/{(4{{{{{{{\rm{\pi }}}}}}}}{n}_{0}{m}_{{{{{{{{\rm{i}}}}}}}}})}^{1/2}$$ based on Bx0 and the background density n0, and the ion and electron inertial length ds ≡ $$c/{(4{{{{{{{\rm{\pi }}}}}}}}{n}_{0}{{{{{{{{\rm{e}}}}}}}}}^{2}/{m}_{{{{{{\mathrm{s}}}}}}})}^{1/2}$$ for species s = i and e, respectively. The ion to electron mass ratio is mi/me = 400 and the background plasma beta is β = 0.01.

### The role of Hall electromagnetic fields

Figure 1a shows the out-of-plane magnetic field By at time 48/Ωci, which is the Hall quadrupole field within the IDR of magnetic reconnection in collisionless electron-ion plasmas55. Importantly, this Hall quadrupole magnetic field By along with the inward-pointing Hall electric field Ez, shown in Fig. 1b, constitute a Poynting vector Sx = −cEzBy/4π in the x-direction. This component diverts the inflowing electromagnetic energy toward the outflow. This is shown by the streamlines of S = cE × B/4π in yellow, which bend in the x-direction significantly before reaching z = 0. These Hall electromagnetic fields arise from the Hall term in the generalized Ohm’s law25,49,50, $${{{{{{{\bf{E}}}}}}}}+{{{{{{{{\bf{V}}}}}}}}}_{{{{{{{{\rm{i}}}}}}}}}\times {{{{{{{\bf{B}}}}}}}}/c={{{{{{{\bf{J}}}}}}}}\times {{{{{{{\bf{B}}}}}}}}/n{{{{{{{\rm{e}}}}}}}}c-\nabla \cdot {{\mathbb{P}}}_{{{{{{{{\rm{e}}}}}}}}}/n{{{{{{{\rm{e}}}}}}}}+({m}_{{{{{{{{\rm{e}}}}}}}}}/{{{{{{{{\rm{e}}}}}}}}}^{2})d\left({{{{{{{\bf{J}}}}}}}}/n\right)/{{{{{\mathrm{d}}}}}}t$$ where d/dt ≡ ∂t − (J/ne)  . The left-hand side (LHS) is the non-ideal electric field that becomes finite when the ion frozen-in condition is violated. Terms on the right-hand side (RHS) contribute to this violation in kinetic plasmas, including the Hall term, the electron pressure divergence term, and the electron inertia term. Figure 1c shows the terms in the out-of-plane (y) component of Ohm’s law in a vertical cut through the x-line; the Hall term (J × B)y/nec (in purple) is the dominant term supporting the reconnection electric field Ey (in red) between the ion inertial scale di and the electron inertial scale de. The Hall term arises because of the decoupling of the relatively immobile ions from the motion of electrons that remain frozen-in to the magnetic fields55. Electrons, the primary current carrier within the IDR (i.e., J −enVe), then drag (both reconnected and not-yet reconnected) magnetic field lines out of the reconnection plane, producing the out-of-plane quadrupolar Hall magnetic field51,53,56,57.

Since the Hall term dominates the electric field EEHall = J × B/nec inside the IDR, then S = − JE 0 per Poynting’s theorem in the steady state. Along the inflow symmetry line (x = 0) toward the x-line magnetic energy B2/8π → 0 since Bx decreases. Also, Bz = 0 and By = 0 (in antiparallel reconnection) due to symmetry. Consequentially, S 0 requires the S streamlines to be diverted to the outflow direction as illustrated in Fig. 2a (also shown in Fig. 1a, b, consistent with the presence of Sx = − cEzBy/4π). Since Poynting flux transports electromagnetic energy, this S streamline pattern implies an energy void centered around the x-line. This pattern introduces the localization to the diffusion region, even in an initially planar current sheet. In contrast, in resistive-MHD, S = − JE$$-\eta {J}_{{{{{{\mathrm{y}}}}}}}^{2}$$ < 0 where η is the resistivity. Thus, as illustrated in Fig. 2b, S streamlines do not need to bend (i.e., Sx 0), instead ending and distributing energy uniformly on the outflow symmetry line (z = 0). This is why the diffusion region in Sweet-Parker reconnection is not localized.

To quantify the degree of localization, we need to estimate the thermal pressure at the x-line. The key is that JE 0 inside the Hall dominated IDR limits the energy conversion to particles and thus also limits the difference in the zz-component of the pressure tensor between the x-line and the far upstream asymptotic region $${{\Delta }}{P}_{{{{{{\mathrm{zz}}}}}}}^{{{{{{{{\rm{xline}}}}}}}}}\equiv {P}_{{{{{{\mathrm{zz}}}}}}}{| }_{{{{{{{{\rm{xline}}}}}}}}}-{P}_{0}$$ (illustrated in Fig. 2a). Given that magnetic pressure B2/8π = 0 at the antiparallel reconnection x-line, if $${{\Delta }}{P}_{{{{{{\mathrm{zz}}}}}}}^{{{{{{{{\rm{xline}}}}}}}}} < {B}_{{{{{{\mathrm{x}}}}}}0}^{2}/8{{{{{{{\rm{\pi }}}}}}}}$$, the reconnecting field bends toward the x-line as it approaches the x-line due to the force-balance condition54 (P + B2/8π) = (B )B/4π. This bending makes the outflow exhausts open out. This fact will be used to develop a first-principles theory of the reconnection rate.

### Non-vanishing kinetic heating within the IDR

Even though total plasma heating is limited within the IDR because JEHall = 0, there is a nonzero energy conversion arising from the convection electric field −Vi × B/c (the gray curve in Fig. 1c) that is critical for modeling the thermal pressure at the x-line. To reveal the kinetic heating process within the IDR along the inflow direction, we show phase space diagrams in Fig. 3. Figure 3a, b show the initial (reduced) distributions f(vz, z) of electrons and ions, respectively. The density profiles are shown by ns/n0 − 1 in pink where s = e and i. The initial denser and hotter populations are visible within the current sheet (z 1di); they balance the magnetic pressure across the initial Harris sheet. At time 48/Ωci, the phase space diagrams of electrons and ions through the x-line show rich structures in Fig. 3c, d, respectively. The pink profiles therein show that the density around the x-line (z = 0) essentially matches the upstream value. On the other hand, while quasi-neutrality remains valid, ions in Fig. 3d form a phase space hole centered around the x-line.

The ion distribution in Fig. 3d arises because the Hall electric field Ez (Fig. 1b) ballistically accelerates ions from both sides of the current sheet toward the x-line. They penetrate across the mid-plane, forming counter-streaming ion beams and thus this phase space hole1,4,58. Note that the acceleration is “ballistic” for inflowing ions because ions are already demagnetized within the IDR and they see the Hall Ez as a DC field. Importantly, this kinetic heating increases the ion zz pressure component above the asymptotic value, as quantified by ΔPizz ≡ Pizz(z) − P0. Kinetically, the Hall Ez arises from charge-separation due to the relative inflow motion between lighter, faster electrons and heavier, slower ions. This Ez then speeds up ions and slows down electrons, which self-regulates its magnitude. Thus, the associated ΔPizz buildup, even though effective, does not dominate the incoming energy budget during reconnection.

We note that the reconnection electric field Ey is ~6 times smaller than the peak Ez in our simulation, and it accelerates both species in the out-of-plane direction (±y for ions and electrons, respectively). Some energy may be imparted into ΔPzz through particle meandering motions (before particles escape to the outflow region), but the ballistic acceleration by Ey is more efficient in imparting its energy into bulk kinetic energy of the current carriers in the y-direction25,59 rather than to thermal energy. Thus, it is not expected to greatly alter ΔPzz and is ignored here.

In Fig. 3e, we show diagonal elements of the pressure tensor of electrons and ions (normalized to $${B}_{{{{{{\mathrm{x}}}}}}0}^{2}/8{{{{{{{\rm{\pi }}}}}}}}$$) through the x-line at time 48/Ωci. For comparison, the initial pressure of each species, which completely balances the upstream magnetic pressure $${B}_{{{{{{\mathrm{x}}}}}}0}^{2}/8{{{{{{{\rm{\pi }}}}}}}}$$, is marked by the gray-dashed horizontal line (note that $${{\mathbb{P}}}_{{{{{{{{\rm{e}}}}}}}}}={{\mathbb{P}}}_{{{{{{{{\rm{i}}}}}}}}}$$ initially for this simulation). In the steady state, the ion ΔPizz (solid green) is slightly reduced while the electron ΔPezz (dotted green) is almost completely depleted. Other significant pressure depletion occurs in ion ΔPixx (solid blue), electron ΔPexx (dotted blue) and electron ΔPeyy (dotted orange). Ion ΔPiyy (solid orange) is roughly half of ΔPizz.

Our interest is in Pzz, since it affects force-balance in the inflow direction. Along the inflow, the ram pressure ($$\mathop{\sum }\nolimits_{{{{{{\mathrm{s}}}}}}}^{{{{{{{{\rm{i}}}}}}}},{{{{{{{\rm{e}}}}}}}}}{n}_{{{{{{\mathrm{s}}}}}}}{m}_{{{{{{\mathrm{s}}}}}}}{V}_{{{{{{\mathrm{sz}}}}}}}^{2}$$) is small, so the force-balance condition in the z-direction is $${({{{{{{{\bf{J}}}}}}}}\times {{{{{{{\bf{B}}}}}}}})}_{{{{{{\mathrm{z}}}}}}}/c\simeq {(\nabla \cdot {\mathbb{P}})}_{{{{{{\mathrm{z}}}}}}}$$, where the total pressure $${\mathbb{P}}\equiv \mathop{\sum }\nolimits_{{{{{{\mathrm{s}}}}}}}^{{{{{{{{\rm{i}}}}}}}},{{{{{{{\rm{e}}}}}}}}}{{\mathbb{P}}}_{{{{{{\mathrm{s}}}}}}}$$. Using J × B/c = BB/4π − B2/8π and integrating along the inflow (z) direction, the force-balance condition reads

$$\frac{{B}^{2}}{8{{{{{{{\rm{\pi }}}}}}}}}+{{\Delta }}{P}_{{{{{{\mathrm{zz}}}}}}}-{\int}^{{{{{{\mathrm{z}}}}}}}\frac{{{{{{{{\bf{B}}}}}}}}\cdot \nabla {B}_{{{{{{\mathrm{z}}}}}}}}{4{{{{{{{\rm{\pi }}}}}}}}}{{{{{\mathrm{dz}}}}}}^{\prime} \simeq {{{{{{{\rm{constant}}}}}}}}.$$
(1)

Initially, the magnetic pressure is totally balanced by the thermal pressure with no contribution from curvature. In the steady-state shown in Fig. 3f, the thermal pressure (green) at the x-line drops significantly, and the upstream magnetic tension (blue) at the mesoscale develops to counter balance the magnetic pressure (red). This tension force is realized through the bending of upstream field lines, which produces the Petschek-type open exhaust with a localized diffusion region31,54.

### Estimating the pressure within the IDR

Within the IDR and outside the EDR, the electric field is E = − Ve × B/c = − Vi × B/c + J × B/nec. Thus the local energy conversion rate is

$${{{{{{{\bf{J}}}}}}}}\cdot {{{{{{{\bf{E}}}}}}}}=-{{{{{{{\bf{J}}}}}}}}\cdot {{{{{{{{\bf{V}}}}}}}}}_{{{{{{{{\rm{i}}}}}}}}}\times {{{{{{{\bf{B}}}}}}}}/c={{{{{{{{\bf{V}}}}}}}}}_{{{{{{{{\rm{i}}}}}}}}}\cdot {{{{{{{\bf{J}}}}}}}}\times {{{{{{{\bf{B}}}}}}}}/c={{{{{{{\rm{e}}}}}}}}n{{{{{{{{\bf{V}}}}}}}}}_{{{{{{{{\rm{i}}}}}}}}}\cdot {{{{{{{{\bf{E}}}}}}}}}_{{{{{{{{\rm{Hall}}}}}}}}}.$$
(2)

All of the work is done on the ions since the rate of work done on electrons is − enVeE = enVe (Ve × B/c) = 0. Note that even though JEHall = 0, the ions do gain energy by EHall, consistent with the ballistic acceleration displayed in Fig. 3d.

Near the inflow symmetry line (x = 0), Jx, By, Bz, and Vix are negligible because of the symmetry in antiparallel reconnection. Thus JE − VizJyBx/c + ViyJzBx/c. The first term is work done by the Hall Ez − JyBx/nec, while the second term is work done by the reconnection electric field EyJzBx/nec. Following the discussion of Fig. 3, the effective ΔPizz buildup primarily arises from ballistic acceleration by the Hall Ez that results in counter-streaming ions near the x-line. We calculate the contribution of−VizJyBx/c to the total rate of energy conversion ∫JEdV for the volume enclosed by the Gaussian surface (1–2–3–4) in Fig. 4a. The right surface (3–4) is chosen to coincide with an ion streamline near the inflow symmetry line, so there is no ion energy flux through that surface. The x-location of this surface relative to x = 0 is quantified by (z). Then

$$\int_{1234}-\frac{{V}_{{{{{{{{\rm{i}}}}}}}}z}{J}_{{{{{{\mathrm{y}}}}}}}{B}_{{{{{{\mathrm{x}}}}}}}}{c}{{{{{\mathrm{d}}}}}}V =\int_{1234}-{V}_{{{{{{{{\rm{i}}}}}}}}z}\frac{c}{4{{{{{{{\rm{\pi }}}}}}}}}\left(\frac{\partial {B}_{{{{{{\mathrm{x}}}}}}}}{\partial z}\right)\frac{{B}_{{{{{{\mathrm{x}}}}}}}}{c}{{{{{\mathrm{d}}}}}}x{{{{{\mathrm{d}}}}}}y{{{{{\mathrm{d}}}}}}z\\ \simeq {\ell }_{{{{{{\mathrm{y}}}}}}}\int\nolimits_{{{{{{{\mathrm{d}}}}}}}_{{{{{{{{\rm{e}}}}}}}}}}^{{d}_{{{{{{{{\rm{i}}}}}}}}}}\ell (z){V}_{{{{{{{{\rm{i}}}}}}}}z}(z)\frac{\partial }{\partial z}\left(\frac{{B}_{{{{{{\mathrm{x}}}}}}}^{2}}{8{{{{{{{\rm{\pi }}}}}}}}}\right){{{{{\mathrm{d}}}}}}z\\ \simeq \left(\frac{{B}_{{{{{{\mathrm{x}}}}}}{{{{{{{\rm{i}}}}}}}}}^{2}-{B}_{{{{{{\mathrm{x}}}}}}{{{{{{{\rm{e}}}}}}}}}^{2}}{8{{{{{{{\rm{\pi }}}}}}}}}\right){V}_{{{{{{{{\rm{i}}}}}}}}z}({d}_{{{{{{{{\rm{i}}}}}}}}})\ell ({d}_{{{{{{{{\rm{i}}}}}}}}}){\ell }_{{{{{{\mathrm{y}}}}}}}.$$
(3)

Here we assumed xBzzBx in writing Jy (c/4π)∂zBx. We take the (z) → 0 limit, so the variables within the integrand are approaximately independent of x. We defined $${B}_{{{{{{\mathrm{x}}}}}}{{{{{{{\rm{i}}}}}}}}}\equiv {B}_{{{{{{\mathrm{x}}}}}}}{| }_{{d}_{{{{{{{{\rm{i}}}}}}}}}}$$ and $${B}_{{{{{{\mathrm{x}}}}}}{{{{{{{\rm{e}}}}}}}}}\equiv {B}_{{{{{{\mathrm{x}}}}}}}{| }_{{{{{{{\mathrm{d}}}}}}}_{{{{{{{{\rm{e}}}}}}}}}}$$, and used y as the dimension of the integrated surface in the out-of-plane (translationally invariant) direction. We also assumed that the density is nearly incompressible along the inflow so Viz(z)(z) Viz(di)(di).

From the discussion of Fig. 3d, we expect that most of this energy is converted to Pizz, which contributes to the ion enthalpy flux3 Hi = $$(1/2)\,{{\mbox{Tr}}}\,({{\mathbb{P}}}_{{{{{{{{\rm{i}}}}}}}}}){{{{{{{{\bf{V}}}}}}}}}_{{{{{{{{\rm{i}}}}}}}}}$$ + $${{\mathbb{P}}}_{{{{{{{{\rm{i}}}}}}}}}\cdot {{{{{{{{\bf{V}}}}}}}}}_{{{{{{{{\rm{i}}}}}}}}}$$ that enters the EDR in the vicinity of the x-line. The associated net enthalpy flux difference  HdA through the Gaussian surface 1–2–3–4 is $$(3/2){P}_{{{{{{{{\rm{i}}}}}}}}{{{{{\mathrm{zz}}}}}}}{| }_{{d}_{{{{{{{{\rm{e}}}}}}}}}}{V}_{{{{{{{{\rm{i}}}}}}}}{{{{{\mathrm{z}}}}}}}({d}_{{{{{{{{\rm{e}}}}}}}}})\ell ({d}_{{{{{{{{\rm{e}}}}}}}}}){\ell }_{{{{{{\mathrm{y}}}}}}}$$ − $$(3/2){P}_{{{{{{{{\rm{i}}}}}}}}{{{{{\mathrm{zz}}}}}}}{| }_{{d}_{{{{{{{{\rm{i}}}}}}}}}}{V}_{{{{{{{{\rm{i}}}}}}}}{{{{{\mathrm{z}}}}}}}({d}_{{{{{{{{\rm{i}}}}}}}}})\ell ({d}_{{{{{{{{\rm{i}}}}}}}}}){\ell }_{{{{{{\mathrm{y}}}}}}}$$$$(3/2){P}_{{{{{{{{\rm{i}}}}}}}}{{{{{\mathrm{zz}}}}}}}{| }_{{d}_{{{{{{{{\rm{i}}}}}}}}}}^{{d}_{{{{{{{{\rm{e}}}}}}}}}}{V}_{{{{{{{{\rm{i}}}}}}}}z}({d}_{{{{{{{{\rm{i}}}}}}}}})\ell ({d}_{{{{{{{{\rm{i}}}}}}}}}){\ell }_{{{{{{\mathrm{y}}}}}}}$$, where $${P}_{{{{{{{{\rm{i}}}}}}}}{{{{{\mathrm{zz}}}}}}}{| }_{{d}_{{{{{{{{\rm{i}}}}}}}}}}^{{d}_{{{{{{{{\rm{e}}}}}}}}}}$$ ≡ $${P}_{{{{{{{{\rm{i}}}}}}}}{{{{{\mathrm{zz}}}}}}}{| }_{{d}_{{{{{{{{\rm{e}}}}}}}}}}$$ − $${P}_{{{{{{{{\rm{i}}}}}}}}{{{{{\mathrm{zz}}}}}}}{| }_{{d}_{{{{{{{{\rm{i}}}}}}}}}}$$. Equating this quantity with the RHS of Eq. (3) gives

$${P}_{{{{{{{{\rm{i}}}}}}}}{{{{{\mathrm{zz}}}}}}}{| }_{{d}_{{{{{{{{\rm{i}}}}}}}}}}^{{d}_{{{{{{{{\rm{e}}}}}}}}}}\simeq \frac{2}{3}\left(\frac{{B}_{{{{{{\mathrm{x}}}}}}{{{{{{{\rm{i}}}}}}}}}^{2}-{B}_{{{{{{\mathrm{x}}}}}}{{{{{{{\rm{e}}}}}}}}}^{2}}{8{{{{{{{\rm{\pi }}}}}}}}}\right).$$
(4)

Since no work is done on electrons outside the EDR, $${P}_{{{{{{{{\rm{e}}}}}}}}{{{{{\mathrm{zz}}}}}}}{| }_{{d}_{{{{{{{{\rm{i}}}}}}}}}}^{{d}_{{{{{{{{\rm{e}}}}}}}}}}$$ 0 and thus the total thermal pressure difference $${P}_{{{{{{\mathrm{zz}}}}}}}{| }_{{d}_{{{{{{{{\rm{i}}}}}}}}}}^{{d}_{{{{{{{{\rm{e}}}}}}}}}}$$$${P}_{{{{{{{{\rm{i}}}}}}}}{{{{{\mathrm{zz}}}}}}}{| }_{{d}_{{{{{{{{\rm{i}}}}}}}}}}^{{d}_{{{{{{{{\rm{e}}}}}}}}}}$$. This thermal pressure increase is smaller than the magnetic pressure drop $$({B}_{{{{{{\mathrm{x}}}}}}{{{{{{{\rm{i}}}}}}}}}^{2}-{B}_{{{{{{\mathrm{x}}}}}}{{{{{{{\rm{e}}}}}}}}}^{2})/8{{{{{{{\rm{\pi }}}}}}}}$$ between the di-scale and de-scale, so there is insufficient pressure to balance forces in the z-direction without the bending of field lines, and Hall reconnection opens into a Petschek-type geometry. This predicted value of Eq. (4), calculated using the measured Bxi and Bxe, is plotted as a horizontal magenta line in Fig. 3e and compares well with the measured $${P}_{{{{{{{{\rm{i}}}}}}}}{{{{{\mathrm{zz}}}}}}}{| }_{{d}_{{{{{{{{\rm{i}}}}}}}}}}^{{d}_{{{{{{{{\rm{e}}}}}}}}}}$$ (green).

### Available magnetic energy at the EDR scale

In order to estimate the relative magnetic pressure (energy) at the EDR, we write

$$\frac{c{E}_{{{{{{\mathrm{y}}}}}}{{{{{{{\rm{i}}}}}}}}}}{{B}_{{{{{{\mathrm{x}}}}}}{{{{{{{\rm{i}}}}}}}}}{V}_{{{{{{{{\rm{Ai}}}}}}}}}}=\frac{{V}_{{{{{{{{\rm{in,i}}}}}}}}}}{{V}_{{{{{{{{\rm{Ai}}}}}}}}}}\simeq \frac{{d}_{{{{{{{{\rm{i}}}}}}}}}}{{L}_{{{{{{{{\rm{i}}}}}}}}}} \sim \frac{{d}_{{{{{{{{\rm{e}}}}}}}}}}{{L}_{{{{{{{{\rm{e}}}}}}}}}}\simeq \frac{{V}_{{{{{{{{\rm{in,e}}}}}}}}}}{{V}_{{{{{{{{\rm{Ae}}}}}}}}}}=\frac{c{E}_{{{{{{\mathrm{y}}}}}}{{{{{{{\rm{e}}}}}}}}}}{{B}_{{{{{{\mathrm{x}}}}}}{{{{{{{\rm{e}}}}}}}}}{V}_{{{{{{{{\rm{Ae}}}}}}}}}}.$$
(5)

The quantities are defined and illustrated in Fig.  4b. The first and last equalities come from the frozen-in conditions Eys = Vin,sBxs/c at the inflow edges of the IDR and EDR for s = i and e, respectively. We use incompressibility for the second and fourth equalities. For the third equality, we use a geometrical argument that the magnetic field line threading the x-line and the corners of the EDR and the IDR is approximately straight, resulting in a similar aspect ratio for the EDR and the IDR. At the ion-scale, the outflow speed is the ion Alfvén speed $${V}_{{{{{{{{\rm{Ai}}}}}}}}}\equiv {B}_{{{{{{\mathrm{x}}}}}}{{{{{{{\rm{i}}}}}}}}}/{(4{{{{{{{\rm{\pi }}}}}}}}n{m}_{{{{{{{{\rm{i}}}}}}}}})}^{1/2}$$. In contrast, the electron outflow speed is the electron Alfvén speed based on the local conditions, $${V}_{{{{{{{{\rm{Ae}}}}}}}}}\equiv {B}_{{{{{{\mathrm{x}}}}}}{{{{{{{\rm{e}}}}}}}}}/{(4{{{{{{{\rm{\pi }}}}}}}}n{m}_{{{{{{{{\rm{e}}}}}}}}})}^{1/2}$$, since ions decouple from the motion of magnetic field lines in the electron-scale inside the IDR20.

By equating the first and last terms and noting that Ey is uniform in 2D steady-state per Faraday’s law (seen in Fig. 1c), we find

$$\frac{{B}_{{{{{{\mathrm{x}}}}}}{{{{{{{\rm{e}}}}}}}}}^{2}}{{B}_{{{{{{\mathrm{x}}}}}}{{{{{{{\rm{i}}}}}}}}}^{2}}\simeq {\left(\frac{{m}_{{{{{{{{\rm{e}}}}}}}}}}{{m}_{{{{{{{{\rm{i}}}}}}}}}}\right)}^{1/2}.$$
(6)

Note that the equality between the first and the last terms is consistent with the high-cadence observation of Magnetospheric Multiscale Mission (MMS)60. For mi/me = 400 as in the simulation, $${B}_{{{{{{\mathrm{x}}}}}}{{{{{{{\rm{e}}}}}}}}}^{2}/{B}_{{{{{{\mathrm{x}}}}}}{{{{{{{\rm{i}}}}}}}}}^{2}\simeq 0.05$$. The predicted $${B}_{{{{{{\mathrm{x}}}}}}{{{{{{{\rm{e}}}}}}}}}^{2}/8{{{{{{{\rm{\pi }}}}}}}}$$ based on the measured $${B}_{{{{{{\mathrm{x}}}}}}{{{{{{{\rm{i}}}}}}}}}^{2}/8{{{{{{{\rm{\pi }}}}}}}}$$ in Fig. 3e compares well with the small $${B}_{{{{{{\mathrm{x}}}}}}}^{2}/8{{{{{{{\rm{\pi }}}}}}}}$$ (B2/8π in black) value at the de-scale. For the real proton to electron mass ratio mi/me = 1836, $${B}_{{{{{{\mathrm{x}}}}}}{{{{{{{\rm{e}}}}}}}}}^{2}/{B}_{{{{{{\mathrm{x}}}}}}{{{{{{{\rm{i}}}}}}}}}^{2}\simeq 0.023$$. The smallness of $${B}_{{{{{{\mathrm{x}}}}}}{{{{{{{\rm{e}}}}}}}}}^{2}/{B}_{{{{{{\mathrm{x}}}}}}{{{{{{{\rm{i}}}}}}}}}^{2}$$ makes the contribution of the pressure depletion within the EDR negligible. However, this imbalanced pressure becomes critical in pair plasmas where the EDR is the same as the IDR, as discussed later.

### Cross-scale coupling and the rate prediction

To predict the reconnection rate, we use the force-balance condition $$\nabla {B}^{2}/8{{{{{{{\rm{\pi }}}}}}}}+\nabla \cdot {\mathbb{P}}={{{{{{{\bf{B}}}}}}}}\cdot \nabla {{{{{{{\bf{B}}}}}}}}/4{{{{{{{\rm{\pi }}}}}}}}$$ and geometry to couple the solutions at the IDR, EDR and the upstream MHD region. First, we discretize this equation at point 7 of Fig. 4c. In the z-direction,

$$\frac{{B}_{{{{{{\mathrm{x}}}}}}{{{{{{{\rm{i}}}}}}}}}^{2}-{B}_{{{{{{\mathrm{x}}}}}}{{{{{{{\rm{e}}}}}}}}}^{2}}{8{{{{{{{\rm{\pi }}}}}}}}({d}_{{{{{{{{\rm{i}}}}}}}}}-{d}_{{{{{{{{\rm{e}}}}}}}}})}-\frac{{P}_{{{{{{\mathrm{zz}}}}}}}{| }_{{d}_{{{{{{{{\rm{i}}}}}}}}}}^{{d}_{{{{{{{{\rm{e}}}}}}}}}}}{{d}_{{{{{{{{\rm{i}}}}}}}}}-{d}_{{{{{{{{\rm{e}}}}}}}}}}\simeq \left(\frac{{B}_{{{{{{\mathrm{x}}}}}}{{{{{{{\rm{i}}}}}}}}}+{B}_{{{{{{\mathrm{x}}}}}}{{{{{{{\rm{e}}}}}}}}}}{2}\right)\frac{2{B}_{{{{{{\mathrm{z}}}}}}8}}{4{{{{{{{\rm{\pi }}}}}}}}{L}_{{{{{{{{\rm{i}}}}}}}}}({d}_{{{{{{{{\rm{i}}}}}}}}}-{d}_{{{{{{{{\rm{e}}}}}}}}})/{d}_{{{{{{{{\rm{i}}}}}}}}}}.$$
(7)

From geometry, the slope of the separatrix Slopedi/LiBz8/[(Bxi + Bxe)/2]. Solving for Slope gives

$${S}_{{{{{{{{\rm{lope}}}}}}}}}^{2}\simeq \frac{{B}_{{{{{{\mathrm{x}}}}}}{{{{{{{\rm{i}}}}}}}}}^{2}-{B}_{{{{{{\mathrm{x}}}}}}{{{{{{{\rm{e}}}}}}}}}^{2}}{{({B}_{{{{{{\mathrm{x}}}}}}{{{{{{{\rm{i}}}}}}}}}+{B}_{{{{{{\mathrm{x}}}}}}{{{{{{{\rm{e}}}}}}}}})}^{2}}-\frac{8{{{{{{{\rm{\pi }}}}}}}}{P}_{{{{{{\mathrm{zz}}}}}}}{| }_{{d}_{{{{{{{{\rm{i}}}}}}}}}}^{{d}_{{{{{{{{\rm{e}}}}}}}}}}}{{({B}_{{{{{{\mathrm{x}}}}}}{{{{{{{\rm{i}}}}}}}}}+{B}_{{{{{{\mathrm{x}}}}}}{{{{{{{\rm{e}}}}}}}}})}^{2}}.$$
(8)

Physically, this expression relates the opening angle of the separatrix to the thermal pressure difference. Plugging in $${P}_{{{{{{\mathrm{zz}}}}}}}{| }_{{d}_{{{{{{{{\rm{i}}}}}}}}}}^{{d}_{{{{{{{{\rm{e}}}}}}}}}}$$ using Eq. (4), we get

$${S}_{{{{{{{{\rm{lope}}}}}}}}}^{2}\simeq \frac{1}{3}\left[\frac{1-({B}_{{{{{{\mathrm{x}}}}}}{{{{{{{\rm{e}}}}}}}}}/{B}_{{{{{{\mathrm{x}}}}}}{{{{{{{\rm{i}}}}}}}}})}{1+({B}_{{{{{{\mathrm{x}}}}}}{{{{{{{\rm{e}}}}}}}}}/{B}_{{{{{{\mathrm{x}}}}}}{{{{{{{\rm{i}}}}}}}}})}\right].$$
(9)

A similar analysis on the force-balance across the EDR gives

$${S}_{{{{{{{{\rm{lope}}}}}}}}}^{2}\simeq 1-\frac{8{{{{{{{\rm{\pi }}}}}}}}{P}_{{{{{{\mathrm{zz}}}}}}}{| }_{{d}_{{{{{{{{\rm{e}}}}}}}}}}^{0}}{{B}_{{{{{{\mathrm{x}}}}}}{{{{{{{\rm{e}}}}}}}}}^{2}},$$
(10)

where the slope of the separatrix is assumed similar inside and outside the EDR because the magnetic tension straightens out the field lines.

We now relate Bxi back to the upstream asymptotic field Bx0 using

$$\frac{{B}_{{{{{{\mathrm{x}}}}}}{{{{{{{\rm{i}}}}}}}}}}{{B}_{{{{{{\mathrm{x}}}}}}0}}\simeq \frac{1-{S}_{{{{{{{{\rm{lope}}}}}}}}}^{2}}{1+{S}_{{{{{{{{\rm{lope}}}}}}}}}^{2}},$$
(11)

which was previously derived using the force-balance condition upstream of the IDR at the mesoscale31. We again assume the slope of the separatrix is similar inside and outside the IDR.

Finally, using Eqs. (4), (10) and (11), we can rewrite the total thermal pressure buildup $${{\Delta }}{P}_{{{{{{\mathrm{zz}}}}}}}^{{{{{{{{\rm{xline}}}}}}}}}\simeq {P}_{{{{{{\mathrm{zz}}}}}}}{| }_{{d}_{{{{{{{{\rm{i}}}}}}}}}}^{{d}_{{{{{{{{\rm{e}}}}}}}}}}+{P}_{{{{{{\mathrm{zz}}}}}}}{| }_{{d}_{{{{{{{{\rm{e}}}}}}}}}}^{0}$$ in terms of the upstream asymptotic magnetic pressure $${B}_{{{{{{\mathrm{x}}}}}}0}^{2}/8{{{{{{{\rm{\pi }}}}}}}}$$ as

$${{\Delta }}{P}_{{{{{{\mathrm{zz}}}}}}}^{{{{{{{{\rm{xline}}}}}}}}}\simeq \left[\frac{2}{3}+\left(\frac{1}{3}-{S}_{{{{{{{{\rm{lope}}}}}}}}}^{2}\right)\frac{{B}_{{{{{{\mathrm{x}}}}}}{{{{{{{\rm{e}}}}}}}}}^{2}}{{B}_{{{{{{\mathrm{x}}}}}}{{{{{{{\rm{i}}}}}}}}}^{2}}\right]{\left(\frac{1-{S}_{{{{{{{{\rm{lope}}}}}}}}}^{2}}{1+{S}_{{{{{{{{\rm{lope}}}}}}}}}^{2}}\right)}^{2}\frac{{B}_{{{{{{\mathrm{x}}}}}}0}^{2}}{8{{{{{{{\rm{\pi }}}}}}}}}.$$
(12)

Using Eqs. (12), (9) and (6), for mi/me = 400 we get $${{\Delta }}{P}_{{{{{{\mathrm{zz}}}}}}}^{{{{{{{{\rm{xline}}}}}}}}}$$$$0.283({B}_{{{{{{\mathrm{x}}}}}}0}^{2}/8{{{{{{{\rm{\pi }}}}}}}})$$. This prediction is within  20% of the simulated $${{\Delta }}{P}_{{{{{{\mathrm{zz}}}}}}}^{{{{{{{{\rm{xline}}}}}}}}}/({B}_{{{{{{\mathrm{x}}}}}}0}^{2}/8{{{{{{{\rm{\pi }}}}}}}})$$ as indicated by the magenta horizontal line in Fig. 3f. The predicted $${{\Delta }}{P}_{{{{{{\mathrm{zz}}}}}}}^{{{{{{{{\rm{xline}}}}}}}}}$$ is considerably less than $${B}_{{{{{{\mathrm{x}}}}}}0}^{2}/8{{{{{{{\rm{\pi }}}}}}}}$$, which from Eq. (1) is consistent with there being an open outflow geometry.

Once the separatrix slope is determined, we can also obtain a first-principles prediction of the reconnection rate using the R − Slope relation in Liu et al.31,

$$R={S}_{{{{{{{{\rm{lope}}}}}}}}}{\left(\frac{1-{S}_{{{{{{{{\rm{lope}}}}}}}}}^{2}}{1+{S}_{{{{{{{{\rm{lope}}}}}}}}}^{2}}\right)}^{2}\sqrt{1-{S}_{{{{{{{{\rm{lope}}}}}}}}}^{2}},$$
(13)

which is 0.157 for mi/me = 1836 and 0.172 for mi/me = 400, consistent with the measured rate on the order of $${{{{{{{\mathcal{O}}}}}}}}(0.1)$$ as shown in Fig. 1c and literature21,52,61.

### Pair plasmas and resistive-MHD

In systems without the Hall effect, such as pair plasmas and resistive-MHD, energy conversion inside the diffusion region is totally different. There is no two-scale structure to the diffusion region. Thus, for the following discussion, we denote the microscopic thickness of the diffusion region as dm (corresponding to di in Fig. 4a) and define in ≡ (dm), Vin ≡ Viz(dm), Bxm ≡ Bx(dm) and the inflowing Poynting vector Sin ≡ Sz(dm). We consider the energy conversion rate integrated over the gray rectangular box (1-5-6-4) in Fig. 4a in the in → 0 limit to illustrate the critical difference compared to Hall reconnection. Near the inflow symmetry line (x = 0) in antiparallel reconnection, Jx = 0, thus JE = JyEy + JzEz. Without the Hall effect, Ez vanishes while Ey is uniform in 2D steady-state, thus

$$\int_{1564}{{{{{{{\bf{J}}}}}}}}\cdot {{{{{{{\bf{E}}}}}}}}{{{{{\mathrm{d}}}}}}V \simeq {\ell }_{{{{{{\mathrm{y}}}}}}}{\ell }_{{{{{{{{\rm{in}}}}}}}}}{E}_{{{{{{\mathrm{y}}}}}}}\int\nolimits_{0}^{{d}_{{{{{{{{\rm{m}}}}}}}}}}\frac{c}{4{{{{{{{\rm{\pi }}}}}}}}}\left(\frac{\partial {B}_{{{{{{\mathrm{x}}}}}}}}{\partial z}\right){{{{{\mathrm{d}}}}}}z\\ =c\frac{{E}_{{{{{{\mathrm{y}}}}}}}{B}_{{{{{{\mathrm{x}}}}}}{{{{{{{\rm{m}}}}}}}}}}{4{{{{{{{\rm{\pi }}}}}}}}}{\ell }_{{{{{{{{\rm{in}}}}}}}}}{\ell }_{{{{{{\mathrm{y}}}}}}}={S}_{{{{{{{{\rm{in}}}}}}}}}{\ell }_{{{{{{{{\rm{in}}}}}}}}}{\ell }_{{{{{{\mathrm{y}}}}}}}.$$
(14)

Here we assumed xBzzBx (i.e., weak localization) again, so Jy (c/4π)∂zBx. Importantly, the last equality of Eq. (14) indicates that the Poynting flux entering the top surface (1–4) is all converted to plasma energy within this narrow rectangular box. In other words, no Poynting flux is diverted to the outflow direction, as illustrated in Fig. 2b. The outflowing enthalpy flux Hx and the bulk flow kinetic energy flux3,7 $${{{{{{{\bf{K}}}}}}}}\equiv \mathop{\sum }\nolimits_{{{{{{\mathrm{s}}}}}}}^{{{{{{{{\rm{i}}}}}}}},{{{{{{{\rm{e}}}}}}}}}(1/2)n{m}_{{{{{{\mathrm{s}}}}}}}{V}_{{{{{{\mathrm{s}}}}}}}^{2}{{{{{{{{\bf{V}}}}}}}}}_{{{{{{\mathrm{s}}}}}}}$$ in the x-direction Kx compete for the inflowing energy. For example, in the low background-β limit where both Hz and Kz at the inflow surface (1–4) are negligible, the energy conversion is Sinin$$\int\nolimits_{6}^{4}{H}_{{{{{{\mathrm{x}}}}}}}{{{{{\mathrm{d}}}}}}z$$ + $$\int\nolimits_{6}^{4}{K}_{{{{{{\mathrm{x}}}}}}}{{{{{\mathrm{d}}}}}}z$$.

In electron-positron (mi = me) pair plasmas23,31,62, near the inflow symmetry line Kx primarily comes from the bulk flow kinetic energy of the current carriers $$\mathop{\sum }\nolimits_{{{{{{\mathrm{s}}}}}}}^{{{{{{{{\rm{i}}}}}}}},{{{{{{{\rm{e}}}}}}}}}(1/2)n{m}_{{{{{{\mathrm{s}}}}}}}{V}_{{{{{{\mathrm{sy}}}}}}}^{2}{V}_{{{{{{\mathrm{sx}}}}}}}$$. In the magnetically-dominated relativistic regime15, to sustain the extreme current density, Kx can be large and it limits the energy available for the enthalpy Hx. This leads to a depleted pressure at the x-line and fast reconnection54. A similar competition could occur in non-relativistic low-β pair plasmas. Without the inward-pointing Hall Ez, the counter-streaming ions that efficiently build up ΔPzz in Hall reconnection are absent. The only other potential source of heating is through the reconnection electric field Ey, which efficiently increases the current carrier drift speed, but not ΔPzz because the acceleration is primarily in the y-direction. Therefore, ΔPzz should be less than $${B}_{{{{{{\mathrm{x}}}}}}0}^{2}/8{{{{{{{\rm{\pi }}}}}}}}$$ and fast reconnection with an open outflow should occur.

For (isotropic) resistive-MHD, Kx near the inflow symmetry line vanishes because the current in MHD is not associated with any kinetic energy. All the inflowing energy is then converted into enthalpy. Using Ey = VinBxm/c in Eq. (14) then Sinin = $$({B}_{{{{{{\mathrm{x}}}}}}{{{{{{{\rm{m}}}}}}}}}^{2}/4{{{{{{{\rm{\pi }}}}}}}}){V}_{{{{{{{{\rm{in}}}}}}}}}{\ell }_{{{{{{{{\rm{in}}}}}}}}}$$$$\int\nolimits_{6}^{4}{H}_{{{{{{\mathrm{x}}}}}}}{{{{{\mathrm{d}}}}}}z$$ = $$(5/2)\int\nolimits_{6}^{4}P{V}_{{{{{{\mathrm{x}}}}}}}{{{{{\mathrm{d}}}}}}z$$ < $$(5/2)P{| }_{6}\int\nolimits_{6}^{4}{V}_{{{{{{\mathrm{x}}}}}}}{{{{{\mathrm{d}}}}}}z$$, where the inequality arises from a reasonable thermal pressure profile that peaks at point 6 on the 4-6 line. In the incompressible limit, $$\int\nolimits_{6}^{4}{V}_{{{{{{\mathrm{x}}}}}}}{{{{{\mathrm{d}}}}}}z$$ = Vinin, so we get P6 > $$(4/5){B}_{x{{{{{{{\rm{m}}}}}}}}}^{2}/8{{{{{{{\rm{\pi }}}}}}}}$$, indicating that a balanced-pressure $$P{| }_{4}^{6}$$ = $${B}_{{{{{{\mathrm{x}}}}}}{{{{{{{\rm{m}}}}}}}}}^{2}/8{{{{{{{\rm{\pi }}}}}}}}$$ becomes possible. Numerical simulations confirms that the thermal pressure at the x-line nearly balances the upstream magnetic pressure in Sweet-Parker reconnection. This implies Slope → 0 from Eq. (10) and explains why Sweet-Parker reconnection is slow; note that the Sweet-Parker theory17,18 itself does not address why the diffusion region length extends to the system size (i.e., Slope → 0), but follows naturally in the present model.

In fact, we can recover the Sweet-Parker scaling using the framework laid out here. Using Ey = ηJyxlineη(c/4π)(Bxm/dm) in Eq. (14), we get

$$\int_{1564}{{{{{{{\bf{J}}}}}}}}\cdot {{{{{{{\bf{E}}}}}}}}{{{{{\mathrm{d}}}}}}V\simeq \eta {\left(\frac{c}{4{{{{{{{\rm{\pi }}}}}}}}}\right)}^{2}\frac{{B}_{{{{{{\mathrm{x}}}}}}{{{{{{{\rm{m}}}}}}}}}^{2}}{{d}_{{{{{{{{\rm{m}}}}}}}}}}{\ell }_{{{{{{{{\rm{in}}}}}}}}}{\ell }_{{{{{{\mathrm{y}}}}}}}.$$
(15)

Since this energy is all converted into the enthalpy $$(5/2){\ell }_{{{{{{\mathrm{y}}}}}}}\int\nolimits_{6}^{4}P{V}_{{{{{{\mathrm{x}}}}}}}{{{{{\mathrm{d}}}}}}z$$$$(5/2)\langle P\rangle {\ell }_{{{{{{\mathrm{y}}}}}}}\int\nolimits_{6}^{4}{V}_{{{{{{\mathrm{x}}}}}}}{{{{{\mathrm{d}}}}}}z$$ (5/2)〈PVininy, we can solve for the average pressure 〈P〉. The pressure buildup can be estimated as

$$P{| }_{{d}_{{{{{{{{\rm{m}}}}}}}}}}^{0}\simeq \langle P\rangle \simeq \frac{4}{5}{\left(\frac{{L}_{{{{{{{{\rm{i}}}}}}}}}}{{d}_{{{{{{{{\rm{m}}}}}}}}}}\right)}^{2}\left(\frac{\eta {c}^{2}}{4{{{{{{{\rm{\pi }}}}}}}}{V}_{{{{{{{{\rm{Ai}}}}}}}}}{L}_{{{{{{{{\rm{i}}}}}}}}}}\right)\frac{{B}_{{{{{{\mathrm{x}}}}}}{{{{{{{\rm{m}}}}}}}}}^{2}}{8{{{{{{{\rm{\pi }}}}}}}}},$$
(16)

where we used plasma continuity VinVAidm/Li over the entire diffusion region. Using this $$P{| }_{{d}_{{{{{{{{\rm{m}}}}}}}}}}^{0}$$ in Eq. (10) though noting Pzz = P since the pressure is isotropic, de = di = dm, Bxe = Bxi = Bxm and using Li/dmSlope as before, we get

$${S}_{{{{{{{{\rm{lope}}}}}}}}}^{2}\simeq 1-\frac{4}{5}{\left(\frac{1}{{S}_{{{{{{{{\rm{lope}}}}}}}}}}\right)}^{2}\left(\frac{\eta {c}^{2}}{4{{{{{{{\rm{\pi }}}}}}}}{V}_{{{{{{{{\rm{Ai}}}}}}}}}{L}_{{{{{{{{\rm{i}}}}}}}}}}\right).$$
(17)

Looking for a solution in the Slope → 0 limit, as justified above, we get the reconnection rate from Eq. (13) to be

$${R}_{{{{{{{{\rm{SP}}}}}}}}}\simeq {S}_{{{{{{{{\rm{lope}}}}}}}}}\simeq \sqrt{\frac{4}{5}\left(\frac{\eta {c}^{2}}{4{{{{{{{\rm{\pi }}}}}}}}{V}_{{{{{{{{\rm{Ai}}}}}}}}}{L}_{{{{{{{{\rm{i}}}}}}}}}}\right)}.$$
(18)

Up to a numerical factor near unity, this result is the Sweet-Parker scaling17,18 of the reconnection rate in uniform resistivity MHD.

To see the compatibility of the model developed here with the result of a spatially localized resistivity in MHD40,41,42,43,63, one needs to retain − ∂xBz in the estimate of Jy. This leads to ∫1564 JEdV < Sininy, instead of them being equal as in Eq. (14). Physically, a finite ∂xBz indicates a localized diffusion region and gives rise to an x-component of the Poynting vector cEyBz/4π. This Sx also diverts the inflowing energy to the outflow direction, analogous to −cEzBy/4π arising from the Hall effect (Fig. 1). Hence, a localized diffusion region with a depleted pressure at the x-line takes place. However, in this case, the diffusion region localization is introduced by hand or some unidentified mechanisms44.

## Conclusions

We have shown that, counter-intuitively, a lower energy conversion rate JE along the inflow toward the x-line makes reconnection faster because the lower pressure requires upstream magnetic field lines to bend to enforce force balance, therefore opening the outflow exhaust. We predict that the high thermal pressure required to get an elongated planar current sheet is not energetically sustainable at the x-line of electron-proton plasmas because JEHall = 0. A significant portion of incoming electromagnetic energy is not transported to the x-line, but is diverted to the outflow by Hall fields. The theory presented here directly links the Hall effect to diffusion region localization, and the linkage is the pressure depletion at the x-line. The predicted pressure drop compares well with simulations of collisionless magnetic reconnection. Through cross-scale coupling between the EDR, IDR and the upstream MHD region, the fast reconnection rate of order $${{{{{{{\mathcal{O}}}}}}}}(0.1)$$ is derived (Eqs. (6, 9, 13) from first-principles for the first time to the best of our knowledge. A closer agreement can be made after considering the reconnection outflow speed reduction by thermal pressure effects38,64,65; a predicted R 0.075 can be read off from the R − Slope relation (similar to Eq. (13)) in Fig. 5c of Li and Liu38 using the same Slope, which is in excellent quantitative agreement with the simulated Ey here in Fig. 1c. In addition, the competition between different forms of energy flux explains why Sweet-Parker reconnection does not have an open exhaust and is slow, while reconnection in electron-positron (pair) plasma is fast. The same theoretical framework recovers the Sweet-Parker scaling law (Eq. 18).

This work is dedicated to explaining the primary localization mechanism for a stable single x-line in collisionless plasmas relevant to magnetospheric, solar and laboratory applications. If a stable single x-line can be realized (i.e., the steadiness is justified by the nearly uniform Ey and negligible ∂t(Jy/n) shown in Fig. 1c), the open outflow geometry can suppress the generation of secondary tearing modes46,66,67 and time-dependent dynamics become less important. However, if the localization of a single x-line is weak, the reconnecting layer is very thin, or pressure-balance across the opened outflow exhausts can not be established54, then secondary tearing modes will be triggered. A cycle of fast generation and ejection of secondary tearing islands provides additional localization mechanism to increase the average reconnection rate54,68,69,70,71,72.

Finally, the Hall effect arises whenever the current sheet thins down to the ion kinetic scale, thus even in a thin 3D current sheet, the argument based on the two key points in the “Introduction” still holds. Large-scale 3D PIC simulations also suggest that broader turbulent current sheets can collapse to thin reconnecting layers in the kinetic scale73. Kinetic reconnecting layers also persist and dominate a current sheet that is filled with self-generated turbulence26,74,75,76,77,78. Importantly, the reconnection process is often quasi-2D in nature. This has been ascertained from the abundance of data from the MMS mission, which has unprecedented spatial and temporal resolutions. In particular, 2D simulations have done an excellent job reproducing detailed reconnection dynamics34,35,36,60,79. Nevertheless, it remains an open question to explore whether fast reconnection can proceed in Nature without eventually forming a dominating kinetic current sheet in three-dimensional plasmas.

## Methods

We carry out 2D PIC simulations of magnetic reconnection in a low background β ≡ $$8{{{{{{{\rm{\pi }}}}}}}}{P}_{0}/{B}_{{{{{{\mathrm{x}}}}}}0}^{2}$$ plasma. The simulations are performed using the VPIC code80, which solves Maxwell’s equations and the relativistic Vlasov equation. The simulation employs the Harris current sheet equilibrium81 that has the initial magnetic profile B = $${B}_{{{{{{\mathrm{x}}}}}}0}\tanh (z/\lambda )\hat{x}$$ and the density profile n = n0sech2(z/λ) + n0; here λ = 1di is the initial half-thickness of the current sheet, where di ≡ $$c/{(4{{{{{{{\rm{\pi }}}}}}}}{n}_{0}{{{{{{{{\rm{e}}}}}}}}}^{2}/{m}_{{{{{{{{\rm{i}}}}}}}}})}^{1/2}$$ is the ion-inertial scale based on the background plasma density n0, which is also the peak density of the current sheet population in our setup; this choice avoids current sheet expansion due to density depletion as has been seen82,83. The simulation size is Lx × Lz = 76.8di × 38.4di that spans the domain [−Lx/2, Lx/2] × [−Lz/2, Lz/2] with nx × nz = 12,288 × 6,144 cells. There are 15 billion macro-particles. The x-direction boundaries are periodic, while the z-direction boundaries are conducting for fields and reflecting for particles. We use the ion-to-electron mass ratio mi/me = 400, the temperature ratio Ti/Te = 1, the plasma β = 0.01 based on the background temperature T0 and background density n0, and ωpece = 4, where the electron plasma frequency $${\omega }_{{{{{{{{\rm{pe}}}}}}}}}={(4{{{{{{{\rm{\pi }}}}}}}}{n}_{0}{{{{{{{{\rm{e}}}}}}}}}^{2}/{m}_{{{{{{{{\rm{e}}}}}}}}})}^{1/2}$$ and the electron cyclotron frequency Ωce = eBx0/mec. These result in the background electron thermal speed $${v}_{{{{{{{{\rm{the}}}}}}}}}\equiv {({T}_{0}/{m}_{{{{{{{{\rm{e}}}}}}}}})}^{1/2}=0.0125c$$, ion thermal speed $${v}_{{{{{{{{\rm{thi}}}}}}}}}\equiv {({T}_{0}/{m}_{{{{{{{{\rm{i}}}}}}}}})}^{1/2}=0.000625c$$ and Alfvén speed $${V}_{{{{{{{{\rm{A0}}}}}}}}}\equiv {B}_{x0}/{(4{{{{{{{\rm{\pi }}}}}}}}{n}_{0}{m}_{{{{{{{{\rm{i}}}}}}}}})}^{1/2}=0.0125c$$, well within the non-relativistic regime for solar and magnetospheric applications. A localized initial magnetic field perturbation of amplitude δBz = 0.03Bx0 is added to induce single x-line reconnection at the center of the simulation domain. To reduce noise in the generalized Ohm’s law analysis, the presented data is time-averaged over an interval 0.085/Ωci, where Ωci = eBx0/mic is the ion cyclotron frequency.